The two-flavor Schwinger model at 50: Solving Coleman's puzzles
Abstract
In his 1976 paper "More about the massive Schwinger model," Coleman introduced 1+1-dimensional Quantum Electrodynamics coupled to two charged massive fermions. By applying Abelian bosonization, he elucidated much of the physics of this two-flavor Schwinger model, but he listed three puzzles at the end of his paper. We present new analytical and numerical calculations to solve Coleman's three puzzles and thereby deepen our understanding of this model. These puzzles pertain to the theory with equal fermion masses at θ= 0 and at θ= π, as well as the size of isospin-breaking effects when the fermion masses are unequal. For the puzzle at θ= π, the solution is related to the structure of the zero-temperature phase diagram arXiv:2305.04437: for equal fermion masses m, the model exhibits spontaneous breaking of charge conjugation symmetry and absence of confinement for any value of the gauge coupling g, so that there is a smooth interpolation from weak to strong coupling. Using two-loop Renormalization Group and integrability methods, we show that the mass gap behaves as ∼ m e−0.111 g2/m2 in the strong coupling regime m≪ g. Our numerical results using the lattice Hamiltonian are in good agreement with this behavior. For the puzzle at θ= 0, the solution is related to a level crossing between two isosinglet particles with different discrete quantum numbers; we demonstrate the necessity of such a crossing by comparing integrability and weak coupling calculations, and we also exhibit the crossing numerically. Finally, we provide a new estimate for the size of isospin-breaking effects caused by different fermion masses at strong coupling.
Related
- Disentangling Expressibility, Symmetry Protection, and Hardware Noise in Variational Quantum Simulation of the Two-Flavor Schwinger Model
- The Origin and Fate of Rapid Negative Modes in Coleman--De Luccia Tunneling
- Resolution of outstanding puzzles in B+ → K+ π+ π- decays
- Quantum Time-Lock Puzzles in the Quantum Random Oracle Model
- Resolving the universality and polarization puzzles in J/ψ hadroproduction
BibTeX
@misc{cuomo2026twoflavor,
title = {{The two-flavor Schwinger model at 50: Solving Coleman's puzzles}},
author = {Gabriel Cuomo and Ross Dempsey and Andrei Katsevich and Igor R. Klebanov and Ilia V. Kochergin and Silviu S. Pufu and Benjamin T. Søgaard},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2605.08042},
eprint = {2605.08042},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.08042},
}
RIS
TY - UNPB
TI - The two-flavor Schwinger model at 50: Solving Coleman's puzzles
AU - Gabriel Cuomo
AU - Ross Dempsey
AU - Andrei Katsevich
AU - Igor R. Klebanov
AU - Ilia V. Kochergin
AU - Silviu S. Pufu
AU - Benjamin T. Søgaard
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2605.08042
UR - https://arxiv.org/abs/2605.08042
AB - In his 1976 paper "More about the massive Schwinger model," Coleman introduced 1+1-dimensional Quantum Electrodynamics coupled to two charged massive fermions. By applying Abelian bosonization, he elucidated much of the physics of this two-flavor Schwinger model, but he listed three puzzles at the end of his paper. We present new analytical and numerical calculations to solve Coleman's three puzzles and thereby deepen our understanding of this model. These puzzles pertain to the theory with equal fermion masses at θ= 0 and at θ= π, as well as the size of isospin-breaking effects when the fermion masses are unequal. For the puzzle at θ= π, the solution is related to the structure of the zero-temperature phase diagram arXiv:2305.04437: for equal fermion masses m, the model exhibits spontaneous breaking of charge conjugation symmetry and absence of confinement for any value of the gauge coupling g, so that there is a smooth interpolation from weak to strong coupling. Using two-loop Renormalization Group and integrability methods, we show that the mass gap behaves as ∼ m e^−0.111 g^2/m^2 in the strong coupling regime m≪ g. Our numerical results using the lattice Hamiltonian are in good agreement with this behavior. For the puzzle at θ= 0, the solution is related to a level crossing between two isosinglet particles with different discrete quantum numbers; we demonstrate the necessity of such a crossing by comparing integrability and weak coupling calculations, and we also exhibit the crossing numerically. Finally, we provide a new estimate for the size of isospin-breaking effects caused by different fermion masses at strong coupling.
ER -